Generally, statisticians use a capital letter to represent a random variable and
a lower-case letter, to represent one of its values. For example,

X represents the random variable X.

P(X) represents the probability of X.

P(X = x) refers to the probability that the random variable X is equal to a
particular value, denoted by x. As an example, P(X = 1) refers to the
probability that the random variable X is equal to 1.

Probability Distributions

An example will make clear the relationship between random variables and
probability distributions. Suppose you flip a coin two times. This simple
statistical experiment can have four possible outcomes: HH, HT, TH, and TT.
Now, let the variable X represent the number of Heads that result from this
experiment. The variable X can take on the values 0, 1, or 2. In this example,
X is a random variable; because its value is determined by the outcome of a
statistical experiment.

A probability distribution is a table or an equation that links
each outcome of a statistical experiment with its probability of occurrence.
Consider the coin flip experiment described above. The table below, which
associates each outcome with its probability, is an example of a probability
distribution.

Number of heads

Probability

0

0.25

1

0.50

2

0.25

The above table represents the probability distribution of the random variable
X.

Cumulative Probability Distributions

A cumulative probability refers to the probability that the
value of a random variable falls within a specified range.

Let us return to the coin flip experiment. If we flip a coin two times, we might
ask: What is the probability that the coin flips would result in one or fewer
heads? The answer would be a cumulative probability. It would be the
probability that the coin flip experiment results in zero heads plus the
probability that the experiment results in one head.

P(X < 1) = P(X = 0) + P(X = 1) = 0.25 + 0.50 = 0.75

Like a probability distribution, a cumulative probability distribution can be
represented by a table or an equation. In the table below, the cumulative
probability refers to the probability than the random variable X is less than
or equal to x.

Number of heads:x

Probability:P(X = x)

Cumulative Probability:P(X < x)

0

0.25

0.25

1

0.50

0.75

2

0.25

1.00

Uniform Probability Distribution

The simplest probability distribution occurs when all of the values of a
random variable occur with equal probability. This probability
distribution is called the uniform distribution.

Uniform Distribution. Suppose the
random variable X can assume k different values. Suppose also that the P(X = xk)
is constant. Then,

P(X = xk) = 1/k

Example 1

Suppose a die is tossed. What is the probability that the die will land on 5?

Solution: When a die is tossed, there are 6 possible outcomes represented
by: S = { 1, 2, 3, 4, 5, 6 }. Each possible outcome is a random variable (X),
and each outcome is equally likely to occur. Thus, we have a uniform
distribution. Therefore, the P(X = 5) = 1/6.

Example 2

Suppose we repeat the dice tossing experiment described in Example 1. This
time, we ask what is the probability that the die will land on a number that is
smaller than 5?